On the space \(\operatorname{Eig}(V)\)  there is a canonical tensor which in the Hermitian case reduces to the quantum geometric tensor (QGT). This QGT was first reported in [1]}, where it was found by looking at infinitesimal distance between states. The physical interpretation of this result, in particular the relevance of the tensor to geometric phases, has been discussed in [2]}. We now show that \(\operatorname{Eig}(V)\)  supports a more general tensor, which relates directly to covariant derivatives. Moreover, this generalized tensor makes no reference to an inner product.
